Action Quantization,Energy Quantization,and Time Parametrization

The additional information within a Hamilton–Jacobi representation of quantum mechanics is extra, in general, to the Schrödinger representation. This additional information specifies the microstate of \(\psi \) that is incorporated into the quantum reduced action, W. Non-physical solutions of the quantum stationary Hamilton–Jacobi equation for energies that are not Hamiltonian eigenvalues are examined to establish Lipschitz continuity of the quantum reduced action and conjugate momentum. Milne quantization renders the eigenvalue J. Eigenvalues J and E mutually imply each other. Jacobi’s theorem generates a microstate-dependent time parametrization \(t-\tau =\partial _E W\) even where energy, E, and action variable, J, are quantized eigenvalues. Substantiating examples are examined in a Hamilton–Jacobi representation including the linear harmonic oscillator numerically and the square well in closed form. Two byproducts are developed. First, the monotonic behavior of W is shown to ease numerical and analytic computations. Second, a Hamilton–Jacobi representation, quantum trajectories, is shown to develop the standard energy quantization formulas of wave mechanics.     

Families of Line-Graphs and Their Quantization

Any directed graph G with N vertices and J edges has an associated line-graph L(G) where the J edges form the vertices of L(G). We show that the non-zero eigenvalues of the adjacency matrices are the same for all graphs of such a family L ⁿ (G). We give necessary and sufficient conditions for a line-graph to be quantisable and demonstrate that the spectra of associated quantum propagators follow the predictions of random matrices under very general conditions. Line-graphs may therefore serve as models to study the semiclassical limit (of large matrix size) of a quantum dynamics on graphs with fixed classical behaviour.     

Weight Function Approach to q-Difference Equations for the q-Hypergeometric Polynomials

In this paper we define a new algebra generated by the difference operators D _q and D _q-1 with two analytic functions (x) and (x). Also, we define an operator M = J ₁ J ₂–J ₃ J ₄ s.t. all q-hypergeometric orthogonal polynomials Y _n(x), x cos(), are eigenfunctions of the operator M with eigenvalues _q [n] _q . The choice of (x) and (x) depend on the weight function of Y _n (x).     

On the spectral function of a hole in a one-dimensional quantum antiferromagnet

We discuss the spectral function of a single hole moving in a one-dimensional quantum antiferromagnet. The latter is described by an anisotropic version of thet-J model, wheret is the hopping matrix element. We introduce two independent coupling parametersJ andJ for the Ising and the transverse part of the Heisenberg exchange. Strong electronic correlations which are incorporated in the model prevent the use of usual diagrammatic techniques for dynamic Green functions based on Wick's theorem. For that reason a new projection technique for general correlation functions in terms of cumulants is used. We consider the case of max. For the case of small transverse coupling relative to the Ising part, we give exact expressions for the one hole correlation function. In the limit of vanishing spin fluctuations our result reduces to earlier calculations of the motion of a hole in a one-dimensional Néel state. However, the inclusion of the spin fluctuations leads to drastic modifications of the spectral function.     

Quantum groups as generalized gauge symmetries in WZNW models. Part II. The quantized model

This is the second part of a paper dealing with the “internal” (gauge) symmetry of the Wess–Zumino–Novikov–Witten (WZNW) model on a compact Lie group G. It contains a systematic exposition, for G = SU(n), of the canonical quantization based on the study of the classical model (performed in the first part) following the quantum group symmetric approach first advocated by L.D. Faddeev and collaborators. The internal symmetry of the quantized model is carried by the chiral WZNW zero modes satisfying quadratic exchange relations and an n-linear determinant condition. For generic values of the deformation parameter the Fock representation of the zero modes’ algebra gives rise to a model space of U _q (sl(n)). The relevant root of unity case is studied in detail for n = 2 when a “restricted” (finite dimensional) quotient quantum group is shown to appear in a natural way. The module structure of the zero modes’ Fock space provides a specific duality with the solutions of the Knizhnik–Zamolodchikov equation for the four point functions of primary fields suggesting the existence of an extended state space of logarithmic CFT type. Combining left and right zero modes (i.e., returning to the 2D model), the rational CFT structure shows up in a setting reminiscent to covariant quantization of gauge theories in which the restricted quantum group plays the role of a generalized gauge symmetry.     

Correlations between eigenvalues of a random matrix

Exact analytical expressions are found for the joint probability distribution functions ofn eigenvalues belonging to a random Hermitian matrix of orderN, wheren is any integer andN. The distribution functions, like those obtained earlier forn=2, involve only trigonometrical functions of the eigenvalue differences.     

Classical limit of the quantized hyperbolic toral automorphisms

The canonical quantization of any hyperbolic symplectomorphismA of the 2-torus yields a periodic unitary operator on aN-dimenional Hilbert space,N=1/h. We prove that this quantum system becomes ergodic and mixing at the classical limit (N,N prime) which can be interchanged with the time-average limit. The recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the Lebesgue measure: the Wigner functions of the eigenstates, supported on the classical periodic orbits, are indeed proved to become uniformly speread in phase space.     

Analytical Solution of the Schrödinger Equation for Makarov Potential with any <Emphasis Type="Italic">ℓ</Emphasis> Angular Momentum

We present the analytical solution of the Schrödinger Equation for the Makarov potential within the framework of the asymptotic iteration method for any n and l quantum numbers. Energy eigenvalues and the corresponding wave functions are calculated. We also obtain the same results for the ring shaped Hartmann potential which is the special form of the non-central Makarov potential.     

Small ħ asymptotics for quantum partition functions associated to particles in external Yang-Mills potentials

To a gauge field on a principalG-bundlePM is associated a sequence of quantum mechanical Hamiltonians, as Planck's constant 0 and a sequence of representations _n ofG is taken. This paper studies the associated quantum partition functions, trace exp (–tH _n ), and produces a complete asymptotic expansion, as 0, =1/n, of which the principal term, proportional to the classical partition function, is the familiar classical limit.Research supported in part by Deutsche Forschungsgemeinschaft and NSF grant NoPhy 81-09011A-01On leave of absence from Freie Universität, BerlinResearch supported by NSF grant MCS 820176A01     

Exact results for a generalized classicalO(n) matrix spin model

A generalizedO(n) matrix version of the classical Heisenberg model, introduced by Fuller and Lenard as a classical limit of a quantum model, is solved exactly in one dimension. The free energy is analytic and the pair correlation functions decay exponentially for all finite temperatures. It is shown, however, that even for a finite number of spins the model has a phase transition in then limit. The transition features a specific heat jump, zero long-range order at all temperatures, and zero correlation length at the critical point. The Curie-Weiss version of the model is also solved exactly and shown to have standard mean-field type behavior for all finiten and to differ from the one-dimensional results in then limit.     

Lie-superalgebraical approach to the second quantization

The Lie superalgebraical properties of the ordinary quantum statistics are discussed. It is indicated that the algebra generated byn pairs of Fermi operator is isomorphic to the classical simple Lie algebraB _n , whereasn pairs of Bose operators generate the simple Lie superalgebraB(0,n). An idea of how one can introduce new classes of creation and annihilation operators that satisfy the second quantization postulates and generate other simple Lie superalgebras is given. The statistics corresponding to the Lie algebraA _n is considered in more details.Invited talk at the Symposium on Mathematical Methods in the Theory of Elementary Particles, Liblice castle, Czechoslovakia, June 18–23, 1978.     

Effects of surface nonlinear interactions on the local critical behavior

The energy spectrum and the wave functions of quantum wells in strong magnetic fields parallel to the potential walls are calculated analytically by means of a new, graph supported method. This Arrow Train Method allows to solve the recurrence relations which originate in the evaluation of eigenvalue determinants of infinite order. The energy eigenvalues for infinite barrier height are computed as a power series in the magnetic fieldB and the center of orbit coordinatez ₀. The power series is evaluated up to the 18th order inB ² for the first four levels and for cyclotron radii comparable to or considerable less than the well width. The corresponding wave functions and the field dependent center of mass shifts are obtained.Work supported in part by the Deutsche Forschungsgemeinschaft     

The classical limit of quantum spin systems

We derive a classical integral representation for the partition function,Z ^Q , of a quantum spin system. With it we can obtain upper and lower bounds to the quantum free energy (or ground state energy) in terms of two classical free energies (or ground state energies). These bounds permit us to prove that when the spin angular momentumJ (but after the thermodynamic limit) the quantum free energy (or ground state energy) is equal to the classical value. In normal cases, our inequality isZ ^C (J)Z ^Q (J)Z ^C (J+1).On leave from the Department of Mathematics, M.I.T., Cambridge, Mass. 02139, USA. Work partially supported by National Science Foundation Grant GP-31674X and by a Guggenheim Memorial Foundation Fellowship.     

On orthogonal and symplectic matrix ensembles

The focus of this paper is on the probability,E (O;J), that a setJ consisting of a finite union of intervals contains no eigenvalues for the finiteN Gaussian Orthogonal (=1) and Gaussian Symplectic (=4) Ensembles and their respective scaling limits both in the bulk and at the edge of the spectrum. We show how these probabilities can be expressed in terms of quantities arising in the corresponding unitary (=2) ensembles. Our most explicit new results concern the distribution of the largest eigenvalue in each of these ensembles. In the edge scaling limit we show that these largest eigenvalue distributions are given in terms of a particular Painlevé II function.     

Singular continuous spectrum for palindromic Schrödinger operators

We give new examples of discrete Schrödinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hullX of the potential is strictly ergodic, then the existence of just one potentialx inX for which the operator has no eigenvalues implies that there is a generic set inX for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such anx is that there is azX that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inX. The class includes well-known substitutions like Fibonacci, Thue-Morse, Period Doubling, binary non-Pisot and ternary non-Pisot. We also show that the operator has no absolutely continuous spectrum for allxX ifX derives from a primitive substitution. For potentials defined by circle maps,x _n =1 _J (₀+n), we show that the operator has purely singular continuous spectrum for a generic subset inX for all irrational and every half-open intervalJ.Work partially supported by NSERC.This material is based upon work supported by the National Science Foundation under Grant No. DMS-91-1715. The Government has certain rights in this material.     

Quasiclassical approximation with the centrifugal potential excluded

We develop a modification of the WKB method (the modified quantization method, or MQM) for finding the radial wave functions. The method is based on excluding the centrifugal potential from the quasiclassical momentum and changing correspondingly the phase in the Bohr-Sommerfeld quantization condition. MQM is used to calculate the asymptotic coefficients at zero and at infinity. We use the examples of power-law and funnel potentials to show that MQM not only dramatically broadens the possibilities of studying the energy spectrum and the wave functions analytically but also ensures accuracy to within a few percent even when one calculates states with a radial quantum number n _r ∼1, provided that the angular momentum l is not too large. We also briefly discuss the possibility of generalizing MQM to the relativistic case (the spinless Salpeter equation). Zh. éksp. Teor. Fiz. 116, 511–525 (August 1999)     

Algebraically Self-Consistent Quasiclassical Approximation on Phase Space

The Wigner–Weyl mapping of quantum operators to classical phase space functions preserves the algebra, when operator multiplication is mapped to the binary * operation. However, this isomorphism is destroyed under the quasiclassical substitution of * with conventional multiplication; consequently, an approximate mapping is required if algebraic relations are to be preserved. Such a mapping is uniquely determined by the fundamental relations of quantum mechanics, as is shown in this paper. The resultant quasiclassical approximation leads to an algebraic derivation of Thomas–Fermi theory, and a new quantization rule which—unlike semiclassical quantization—is non-invariant under action transformations of the Hamiltonian, in the same qualitative manner as the true eigenvalues. The quasiclassical eigenvalues are shown to be significantly more accurate than the corresponding semiclassical values, for a variety of 1D and 2D systems. In addition, certain standard refinements of semiclassical theory are shown to be easily incorporated into the quasiclassical formalism.     

A closed-form solution of the Schroedinger equation describing the harmonic oscillator with time dependent frequency and acted on by a time dependent perturbative force

A new form of the semiclassical quantum conditions in non-separable systems is proposed. In two dimensions (2D) it has the form (? = 1)

where C_Σ is the path of a classical trajectory closed in phase space, N_x and N_y are the number of circuits in the x and y ‘senses’ on the invariant toroid and J_x and J_y are the ‘good’ action variables on the toroid; these action variables, J_x and J_y , must have the values 2π(n ₁ + ½) and 2π(n ₂ + ½) respectively where n ₁ and n ₂ are the integer quantum numbers. Closed classical trajectories occur only for the exceptional toroids with rational frequency ratios. In the general case we imply that the trajectory has closed on itself to some arbitrary accuracy. Results for the 2D potentials studied are in agreement with previously published work. It is shown how the method may be extended to 3D systems.     

Mott transition and sign problem for a model of lattice fermions

We consider a model of spinless fermions on the square latticeZ ² with an interaction potential of strengthU>0 at distance one and strengthJ at distance two, in the largeU limit |t|, |J|U, wheret is the hopping amplitude. As the chemical potential is varied, ift=T=0 we find three different phases corresponding to full, half and zero filling fractions. We study the system at low temperatureT0 by a method involving a canonical transformation and a functional integral representation. IfT=0 we locate the phase boundaries of the Mott metal-insulator transition for all |J|U with upper and lower bounds, show that mean field theory is valid ifJ<0 but fails forJ=0 when also the Peierls condition is violated. This result is a quantum extension of the Pirogov-Sinai theory of phase transitions. IfT>0 we have only one sided bounds for the phase boundaries and we can't validate mean field theory in caseJ<0. We introduce a new resummation scheme for low temperature expansions which yields finite and convergent perturbation series and permits us to study issues like the sign problem. Our algorithm gives an optimal canonical transformation for the functional integral such that the expectation of the sign observableS is exp , whereV is the volume and =T ^–1.Partially supported by the Ambrose Monell Foundation during a visit to the Institute for Advanced study.     

Constant-cutoff approach toSU(3)-symmetry breaking for strange dibaryon states

We suggest a quantum stabilization method for theSU(2)-model, based on the constant-cutoff limit of the cutoff quantization method developed by Balakrishnaet al., which avoids the difficulties with the usual soliton boundary conditions pointed out by Iwasaki and Ohyama. We investigate the baryon numberB = 1 sector of the model and show that after the collective coordinate quantization it admits a stable soliton solution which depends on a single dimensional arbitrary constant. We then show that the approach toSU(3)-symmetry breaking for strange dibaryon states proposed by Kopeliovichet al. can be simplified by omitting the Skyrme stabilizing term and using the constant-cutoff stabilization method. We derive the results for spectra of some strange and nonstrange dibaryon states and obtain the numerical results for the absolute masses of these states, in reasonable agreement with the values obtained, using the complete Skyrme model, by Kopeliovichet al.